Some properties of Fibonacci numbers, Fibonacci octonions and generalized Fibonacci-Lucas octonions

In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.

Proposition 2.2. ( [Fl,Sa;14], [Fl,Sa,Io;13]) Let (f n ) n≥0 be the Fibonacci sequence and let (l n ) n≥0 be the Lucas sequence. Then: In the following proposition, we will give other properties of the Fibonacci and Lucas numbers, which will be necessary in the next proofs.
Proposition 2.3. Let (f n ) n≥0 be the Fibonacci sequence and (l n ) n≥0 be the Lucas sequence Then: Proof. i) Using Proposition 2.1 (i) we have: From Proposition 2.2 (ii) and Proposition 2.1 (i), we obtain: ii) Applying Proposition 2.1 (ii), we have: Using Proposition 2.2 (ii,iv), we have: From Proposition 2.1 (i) and Fibonacci recurrence, we obtain: Using Proposition 2.2 (i), we obtain:
Proposition 3.3. Let a be a real number and let F n be the n-th generalized Fibonacci octonion. Then the norm of F n in the generalized octonion algebra O R (a + 1, 2a + 1, 3a + 1, ) is: Using [Ke, Ak; 15] (p.3), we have (3.2) Applying Proposition 2.1 (iii) and Proposition 2.1 (i), we have: From Proposition 2.2 (ii), Proposition 2.3 (ii) and Proposition 2.1 (i), we have: Using several times the recurrence of Fibonacci sequence and Proposition 2.1 (vii), we obtain: Applying Proposition 2.1 (iii) and Proposition 2.1 (vi,i), we have: From Proposition 2.3 (iii) and the recurrence of Fibonacci sequence, we have: Therefore, we obtained that: (3.4) From Proposition 2.1 (vi,i), we have: Applying Proposition 2.3 (iii) and the recurrence of Fibonacci sequence many times, we have: Therefore, we get : We obtain immediately the following remark: Remark 3.1. If a is a real number, a < −1, then, the generalized octonion algebra O R (a + 1, 2a + 1, 3a + 1) is a split algebra.

Generalized Fibonacci-Lucas octonions
In the paper [Fl,Sa;15 (a)], we introduced the generalized Fibonacci -Lucas numbers, namely: if n is an arbitrary positive integer and p, q be two arbitrary integers, the sequence (g n ) n≥1 , where g n+1 = pf n + ql n+1 , n ≥ 0 is called the sequence of the generalized Fibonacci-Lucas numbers. For not make confusions, we will use the notation g p,q n instead of g n . Let O Q (α, β, γ) be the generalized octonion algebra over Q with the basis {1, e 1 , e 2 , ..., e 7 } . We define the n-th generalized Fibonacci-Lucas octonion to be the element of the form G p,q n = g p,q n ·1+g p,q n+1 ·e 1 +g p,q n+2 ·e 2 +g p,q n+3 ·e 3 +g p,q n+4 ·e 4 +g p,q n+5 ·e 5 +g p,q n+6 ·e 6 +g p,q n+7 ·e 7 .
We wonder what algebraic structure determine the generalized Fibonacci-Lucas octonions. First, we make the following remark.
Remark 4.1. Let n, p, q three arbitrary positive integers, p, q ≥ 0. Then, the n-th generalized Fibonacci-Lucas octonion G p,q n = 0 if and only if p = q = 0.
Moreover, applying Remark 4.1, it results that 0∈A. These implies that A is a Z -submodule of the generalized octonions algebra O Q (α, β, γ) .. Since {1, e 1 , e 2 , ..., e 7 } is a basis of A, it results that A is a free Z-module of rank 8.
(ii) From Remark 4.2 (ii), it results immediately that 5G p,q m · 5G p ′ ,q ′ n ∈B, (∀) m, n∈N * , p, q, p ′ , q ′ ∈Z. Using this fact and a similar reason that in the proof of (i), it results that B is a unitary non-associative subalgebra of the generalized octonions algebra O Q (α, β, γ) .